A280534 Number of partitions of n into two parts with the smaller part prime and the larger part squarefree.

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 1, 5, 3, 5, 2, 5, 2, 3, 2, 4, 4, 5, 3, 6, 4, 5, 4, 7, 5, 6, 4, 6, 5, 7, 3, 7, 6, 6, 3, 6, 5, 7, 3, 6, 4, 8, 4, 9, 4, 8, 4, 10, 5, 8, 3, 8, 6, 9, 4, 10, 5, 9, 6, 10, 5, 9, 5, 9, 6, 9, 5, 12, 6, 8, 6, 11, 8

Offset

1,8

Comments

Number of distinct rectangles with squarefree length and prime width such that L + W = n, W <= L. For example, a(16) = 3; the rectangles are 2 X 14, 3 X 13 and 5 X 11. - Wesley Ivan Hurt, Nov 04 2017

Links

Table of n, a(n) for n=1..89.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor(n/2)} c(i) * mu(n-i)^2, where mu is the Möbius function (A008683) and c = A010051.

Maple

with(numtheory): A280534:=n->add(mobius(n-i)^2*(pi(i)-pi(i-1)), i=1..floor(n/2)): seq(A280534(n), n=1..100); # Wesley Ivan Hurt, Jan 04 2017

Mathematica

Table[Sum[MoebiusMu[n - k]^2 * (PrimePi[k] - PrimePi[k - 1]), {k, 1, Floor[n/2]}], {n, 1, 50}] (* G. C. Greubel, Jan 05 2017 *)
Table[Count[IntegerPartitions[n, {2}], _?(SquareFreeQ[#[[1]]]&&PrimeQ[ #[[2]]]&)], {n, 90}] (* Harvey P. Dale, Oct 17 2021 *)

Prog

(PARI) for(n=1, 50, print1(sum(k=1, floor(n/2), isprime(k)*(moebius(n-k))^2), ", ")) \\ G. C. Greubel, Jan 05 2017

Crossrefs

Cf. A008683, A010051, A280535.

Sequence in context: A348172 A319506 A037809 * A129451 A097195 A274138

Adjacent sequences: A280531 A280532 A280533 * A280535 A280536 A280537

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 04 2017

Status

approved